### Cagniard--de Hoop path perturbations with applications to
non-geometric wave arrivals

Non-geometric wave arrivals are often important in seismology and
elastic wave studies related to the non-destructive evaluation of
structures. In particular tunnelling signals caused by
significant differences in the material parameters, and wavespeeds at
interfaces, generate large responses that may often be dominant. This is
common in elastic wave propagation, for instance, when
a source is close to the interface of a faster medium with a slower
medium, the response in the slower medium is dominated by a signal that
has tunnelled through the faster medium. Other instances of
tunnelling
occur when a compressional source is close to a free surface. In this case
the compressional to shear wave conversion at the surface, and the
mismatch between compressional and shear wavespeeds, leads to a sharp
non-geometric shear wave arrival. Equally, thin high velocity layers
demonstrate tunnelling effects that are perturbations of the response
brought about by a source in a surrounding slower medium. In the above
close refers to the viewpoint of an observer some distance away.
In all of the
instances there is a common feature, namely, each
problem contains a ratio of length
scales, x/h, with h either the source depth or layer thickness and
x the observer distance; this ratio of length
scales characterises
the non-geometric responses. Typically, the non-geometric
response arises when the
current problem is a perturbation away from one where the associated
arrival has a direct geometric interpretation.
Such problems are ideally suited to analysis by the Cagniard-de Hoop
technique. Each tunnelling response is identified as a perturbation
away from an exact solution; this leads to highly accurate and relatively
simple explicit asymptotic solutions. The perturbation scheme is
demonstrated here via the solution of two problems: a compressional source
beneath a fluid/solid interface and beneath a thin high velocity layer.
The first problem has separate non-geometric responses due to both the
material mismatch and the wave conversion at the interface. The thin high
velocity layer perturbs the field generated by a compressional source in a
slower surrounding medium. In both cases the non-geometric arrivals are
analysed in detail.

Co-authored with Duncan Williams.

#### To appear J. Engng. Maths

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